2009 HMMT November Guts Round Problem 31

There are two buildings facing each other, each 5 stories high. How many ways can Kevin string ziplines between the buildings so that:

(a) each zipline starts and ends in the middle of a floor.

(b) ziplines can go up, stay flat, or go down, but can't touch each other (this includes touching at their endpoints).

Note that you can't string a zipline between two floors of the same building.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2009 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.